Theorem rnun 5078

Description: Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Assertion

Ref	Expression
rnun	|- ran ( A u. B ) = ( ran A u. ran B )

Proof of Theorem rnun

Step	Hyp	Ref	Expression
1		cnvun 5075	. . . 4 |- `' ( A u. B ) = ( `' A u. `' B )
2	1	dmeqi 4867	. . 3 |- dom `' ( A u. B ) = dom ( `' A u. `' B )
3		dmun 4873	. . 3 |- dom ( `' A u. `' B ) = ( dom `' A u. dom `' B )
4	2, 3	eqtri 2217	. 2 |- dom `' ( A u. B ) = ( dom `' A u. dom `' B )
5		df-rn 4674	. 2 |- ran ( A u. B ) = dom `' ( A u. B )
6		df-rn 4674	. . 3 |- ran A = dom `' A
7		df-rn 4674	. . 3 |- ran B = dom `' B
8	6, 7	uneq12i 3315	. 2 |- ( ran A u. ran B ) = ( dom `' A u. dom `' B )
9	4, 5, 8	3eqtr4i 2227	1 |- ran ( A u. B ) = ( ran A u. ran B )

Syntax hints:   = wceq 1364   u. cun 3155   `'ccnv 4662   dom cdm 4663   ran crn 4664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-cnv 4671  df-dm 4673  df-rn 4674

This theorem is referenced by:  imaundi  5082  imaundir  5083  rnpropg  5149  fun  5430  foun  5523  fpr  5744  fprg  5745  sbthlemi6  7028  exmidfodomrlemim  7268